A new prolongator for multigrid for the curlcurl equation

Tim Boonen

Katholieke Universiteit Leuven, Computer Science Department

Stefan Vandewalle


Abstract

We consider algebraic multigrid methods for the numerical solution of curlcurl systems in computational electromagnetics. Existing prolongation schemes for the curlcurl equation are often based on the Reitzinger-Schöberl prolongator presented in [1]. There, a piecewise constant edge prolongator Pe is derived from a piecewise constant nodal prolongator Pn, such that the commutation property is satisfied. This prolongation scheme can be improved by applying ideas of smoothed aggregation multigrid to it ([2], [3]).

We will present an alternative prolongation scheme that takes as a starting point an arbitrary partition of unity nodal prolongator Pn. We will show that it is possible to associate with the set of coarse nodal elements (the columns of Pn) a set of coarse edge elements. The link between both sets is the analytical formula for edge elements on triangular/tetrahedral meshes

Eij = Ni grad(Nj) - Nj grad(Ni)

We will show that in the coarse setting, this formula has an exact discrete counterpart, which satisfies the commutation property. This prolongation schema contains the Reitzinger-Schöberl edge prolongator as the special case for a piecewise constant nodal prolongator Pn.


[1] S. Reitzinger, J. Schöberl, An algebraic multigrid method for finite element discretizations with edge elements, Numer. Linear Algebra Appl. 2002, 9, p.223-238.
[2] P.B.Bochev, C.J.Garasi, J.J.Hu, A.C.Robinson, R.S.Tuminaro, An improved algebraic multigrid method for solving Maxwell's equations, Siam Journal on Scientific Computing, 25, p.623-642, 2003.
[3] J.Hu, R.Tuminaro, P.Bochev, C.Garasi, A.Robinson, Toward an h-independent Algebraic Multigrid Method for Maxwell's Equations, To appear in Siam Journal on Scientific Computing, 2005.